Unit Circle Coordinates and Key Angles

Unit Circle Coordinates

• Key coordinates on the unit circle include points like (-1/2, √3/2) and signify angles in radians and degrees.
• The point (-√2/2, √2/2) corresponds to 135° or 3π/4 radians, relating to key trigonometric ratios.
• The coordinate (-√3/2, 1/2) relates to 150°, aligning with various sine and cosine values.

Significant Points

• (-1, 0) represents the leftmost point on the unit circle, aligning with 180° or π radians.
• The coordinate (0, -1) indicates the bottom-most point, corresponding to 270° or 3π/2 radians.
• (1, 0) is the rightmost point, indicating 0° or 0 radians.

Quadrantal Angles

• Quadrantal angles include 0°, 90°, 180°, and 270°, representing significant rotations around the circle.
• Each quadrantal angle has distinct coordinates that are critical for understanding trigonometric properties.

Key Angles and Their Values

• For example, 30° or π/6 radians corresponds to the coordinate (√3/2, 1/2), essential for applications in trigonometry.
• The angle 120° or 2π/3 radians relates to (-1/2, √3/2), demonstrating symmetry in the circle.
• Angles such as 225° (5π/4 radians) yield coordinates (-√2/2, -√2/2), important for quadrant IV analysis.

Conversion Between Degrees and Radians

• Understanding the conversion between degrees and radians is crucial; for instance, multiples of π/6, π/4, and π/3 are common references.
• For example, 315° equates to 7π/4 radians, which yields coordinates (√2/2, -√2/2).

Full Circle Rotation

• The concept of a complete rotation is represented by 360° or 2π radians, returning to the starting point (1, 0).
• Each division of the circle by 30° or multiples highlights foundational sine and cosine values.

Angles Beyond 180°

• Angles such as 210° (7π/6) and 240° (4π/3) further expose the values of trigonometric functions beyond half a rotation.
• The coordinates for 300° (5π/3 radians) are (1/2, -√3/2), providing insight into the negative values in specific quadrants.

Summary of Important Angles

• Notable angles with associated coordinates include:
  • 90° (π/2): (0, 1)
  • 135° (3π/4): (-√2/2, √2/2)
  • 150° (5π/6): (-√3/2, 1/2)
  • 270° (3π/2): (0, -1)
  • 330° (11π/6): (√3/2, -1/2)